On solving 2D weakly singular Volterra integral equations of the second kind

In this paper, we present a semi-analytical approach to solving two-dimensional Volterra integral equations of the second kind which include weakly singular kernels. This approach is based on two-dimensional Laplace transform expansion method and on bivariate rational approximants. More specifically, the proposed approach consists in expanding the two-dimensional Laplace transform of the unknown function for large values and inverting it term by term in order to obtain a double convergent series expansion of the solution. Then, using a few coefficients from the obtained convergent series expansion, we provide the bivariate rational function representation of the unknown function of these integral equations through the application of bivariate homogeneous Pad & eacute; approximants. Furthermore, several numerical examples have been provided in order to show the efficiency of our study.